Fact B.3 (Lemma 1 of [1]}) 

Let \(Q_{\tilde{\mu }}\)  be multivariate Gaussian such that \(Q_{\tilde{\mu }}(\mu , \Sigma ) = N \left( \mu , \Sigma \right)\) .
Then if \(R \sim Q_{\tilde{\mu }}(0, I_d)\) , we know that
\( \forall \beta \in (0,1]: \operatornamewithlimits{\mathbb {P}}\left( \Vert R \Vert _2 > \sqrt{d + \sqrt{d \log (1/\beta )} + 2 \log (1/\beta )} \right) \leqslant \beta . \) 

